Concentration points on two and three dimensional modular hyperbolas and applications

Abstract

Let $p$ be a large prime number, $K,L,M,\lambda$ be integers with $1\le M\le p$ and $\gcd (\lambda ,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M;K,L)$ of solutions of the congruence $$xy\equiv \lambda (\mathrm{mod}p),K+1\le x\le K+M,L+1\le y\le L+M$$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\equiv \lambda (\mathrm{mod}p),L+1\le x,y,z\le L+M.$$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\le M^{o(1)}.$ For $I_3(M;L)$ we prove that if $M<p^{1/8}$ then $I_3(M;L)\le M^{o(1)}.$ Our results have applications to some other problems as well. For instance, it follows that if ${\mathcal{I}}_1,{\mathcal{I}}_2,{\mathcal{I}}_3$ are intervals in $\F^*_p$ of length $|{\mathcal{I}}_i|<p^{1/8},$ then $$|{\mathcal{I}}_1\cdot {\mathcal{I}}_2\cdot {\mathcal{I}}_3|=(|{\mathcal{I}}_1|\cdot |{\mathcal{I}}_2|\cdot |{\mathcal{I}}_3|{)}^{1-o(1)}.$$

12 pages

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